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Volatility of Volatility and Leverage Effect from Options
{\em JEL Classification:} C14, C22, C58, G12.
{\em Keywords:} Characteristic function; leverage effect; nonparametric estimation; options; volatility of volatility.
Time-varying volatility is a ubiquitous feature of asset prices. Given the importance of volatility risk for investors, there is a lot of interest in trading derivative products written on the market volatility index VIX. Volatility of volatility is a major source of risk for traders of volatility derivatives. Related to that, there is a growing literature in finance that documents the emergence of volatility of volatility as a separate risk factor and its weak correlation with volatility risk, see e.g., agarwal2017volatility, hollstein2018aggregate, huang2019volatility and chen2022volatility among others.
Another stylized feature of volatility is its tendency to move in the opposite direction to the asset price. Following black1976studies, this negative covariance/correlation between asset return and asset volatility is referred to as leverage effect. The economic origins of the leverage effect have been extensively studied in earlier work in finance, see e.g., black1976studies, christie1982stochastic, french1987expected, campbell1992no, engle1993measuring, bekaert2000asymmetric, bollerslev2012volatility and bollerslev2023jump, among others.\footnote{Alternative discrete-time stochastic volatility models with leverage effect have been estimated by yu2005leverage.}
Since stochastic volatility is not directly observed, estimating the spot volatility of volatility and the spot leverage effect from asset returns is rather challenging. In this paper, we propose nonparametric estimators for these quantities using options written on the underlying asset. The proposed estimators can be used as diagnostic tools for stochastic volatility modeling. They can be also used for studying the asset pricing implications of volatility of volatility risk and for separating alternative economic explanations regarding the role of volatility risk in equilibrium asset pricing models as done in some of the work mentioned in the previous paragraph.
One way to estimate the volatility of volatility and the leverage effect is to use high-frequency asset return data. More specifically, one can first construct local estimators of volatility and then form realized variance of variance and realized covariance between price and variance from these estimates, see e.g., vetter2015estimation, sanfelici2015high, clinet2021estimation, li2022volatility and toscano2022volatility for volatility of volatility and WM14, AFLWY17, KX17 and yang2023estimation for the leverage effect. The resulting estimators of integrated volatility of volatility and leverage effect, however, have a relatively slow rate of convergence. This reflects the difficulty of such an estimation problem. More specifically, if the asset prices do not contain market microstructure noise, the best attainable rate for integrated volatility of volatility and leverage effect is $n^{1/4}$, where $n$ is the number of high-frequency increments in the fixed time interval. If asset prices are contaminated with noise, this rate drops to $n^{1/8}$. Furthermore, if one is interested in spot instead of integrated volatility of volatility and leverage effect, then the above rates drop by another factor of one half. In addition, the results for the return-based volatility of volatility estimators are derived under various nontrivial restrictions for the underlying asset price, e.g., vetter2015estimation does not allow for price and volatility jumps while li2022volatility rule out volatility jumps.
This paper proposes an alternative approach for volatility of volatility and leverage effect estimation that is based on short-dated options, i.e., options with very short time-to-maturity.\footnote{Options written on different underlying assets and expiring within a few weeks are actively traded on exchanges. This is facilitated by option exchanges offering the so-called weekly options, i.e., options that expire on a weekly basis, with the shortest ones expiring in a week from the issue date. Starting in 2022, the CBOE options exchange offers options on the S&P 500 index that expire at the end of each trading day. In 2022, more than 60% of the trading volume of S&P 500 index options was in options expiring within 7 calendar days.} The idea is to replace the local estimates of spot volatility from high-frequency returns with ones formed from short-dated options. As is well known, short-dated options can render the latent volatility directly observable and that makes the problem of nonparametric estimation of volatility of volatility and the leverage effect significantly easier.\footnote{For recovery of the spot volatility in a model-free way, we need the time-to-maturity of the options to be shrinking asymptotically.} Indeed, todorov2022information document nontrivial gains from using option data for spot volatility estimation. This advantage naturally caries over to the case of estimating volatility of volatility and the leverage effect.\footnote{BFR23,BFR23b also use short-dated options data to estimate the spot volatility of volatility and the spot leverage effect (among other asset price characteristics), without a proof of consistency or asymptotic normality of these estimators. This is done in a semiparametric setting in which price jumps are modeled parametrically and there are no volatility jumps. Our procedure, by contrast, is fully nonparametric and allows for volatility jumps (either positive or negative) in particular.}
In our inference procedures we use the spot volatility estimator of T19, which is formed from option-based nonparametric estimates of the conditional characteristic function of price increments. T19 shows that, for any fixed $u>0$, $\mathbb{E}_t[e^{iu(x_{t+T}-x_t)/\sqrt{T}}]$ can be used to recover nonparametrically the spot diffusive volatility of a general It\^o semimartingale process $x$ when $T$ is asymptotically shrinking.\footnote{In our application, we use $T$ of up to $16$ business days and the median highest $T$ on each day in our sample is only $6$ business days.} As usual, $\mathbb{E}_t$ in the above denotes conditional expectation given information up to time $t$. In CT23_a, we derive a higher order expansion for $\mathbb{E}_t[e^{iu(x_{t+T}-x_t)/\sqrt{T}}]- \mathbb{E}_{t-\Delta}[e^{iu(x_{t+T}-x_{t-\Delta})/\sqrt{T+\Delta}}]$ when both $\Delta$ and $T$, with $0<\Delta<T$, are asymptotically shrinking. We use this result here to quantify the difference between the high-frequency increments of our option-based volatility estimator and the infeasible high-frequency increments of the latent spot diffusive volatility. This, in turn, can be used to estimate nonparametrically various quantities associated with the volatility dynamics, and in particular the volatility of volatility and the leverage effect.
Our volatility of volatility estimator is based on sample variance and first-order autocovariance of the option-based spot volatility increments. The reason for using autocovariance in the estimation is to account for the effect of option observation errors. Option observation errors can be viewed as the natural counterpart to the market microstructure noise in the underlying asset price. Their presence introduces an upward bias in the sample variance of the volatility returns. The first-order autocovariance of the variance return estimates can correct for it. Turning next to the leverage effect, its estimator is constructed in an analogous way from the sample covariance between the price and the estimated volatility increments.
We show consistency of the two estimators and derive a Central Limit Theorem (CLT) for them. The limit distribution is determined by two sources of error in the estimation procedure. One is due to the diffusive innovations in the true (latent) volatility process. The other is due to the observation errors in the options with strikes in the vicinity of the current stock price. The size of the two errors is governed by the time gap between observations for the first one and the tenor and the strike gaps for the second one. All of these quantities are asymptotically shrinking and for our estimation procedure we do not require a condition on their relative size.
The performance of the proposed estimators is evaluated on simulated data from an asset pricing model calibrated to match key features of observed stock and option data. Our Monte Carlo shows that the option-based estimators of volatility of volatility and the leverage effect are significantly more efficient than their return-based counterparts. This is the case in spite of the fact that the latter are constructed using a much higher sampling frequency than the option based ones (five seconds versus one minute). In an empirical application, we compute the volatility of volatility and the leverage effect of the S&P 500 market index using short-dated options written on the index. We use a log transform of the volatility when computing these quantities, i.e., our interest is volatility of log-volatility and covariance between log-price and log-volatility. Our nonparametric results show that volatility and volatility of volatility exhibit only weak dependence while the leverage effect estimates and the market variance are inversely related.
On a theoretical level, the papers closest to the current work are andersen2015exploring and KX17. These papers propose volatility of volatility and leverage effect estimators using the VIX volatility index instead of the nonparametric spot volatility estimators employed here. There are two major differences between the estimators of these papers and ours. First, the VIX index is an estimate of the conditional risk-neutral expectation of one month ahead return quadratic variation. It has dynamics which is different from that of the latent spot diffusive volatility. Hence, the estimands of andersen2015exploring and KX17 on one hand and of our paper on the other hand are different. Second, andersen2015exploring and KX17 do not allow for option measurement error while we do. In fact, both the construction of our estimators and their limiting distributions are impacted by the presence of option observation errors.
The rest of the paper is organized as follows. In Section (ref), we introduce our setting and define the objects of interest in the paper. Section (ref) describes the option observation scheme and the construction of volatility estimators from options. Our estimators and the asymptotic results for them are given in Section (ref). Section (ref) contains Monte Carlo evidence and Section (ref) our empirical application. Section (ref) concludes. Some asymptotic expansion formulas for volatility estimators based on characteristic functions are stated in Appendix (ref), while the proofs of the theoretical results of this paper are given in Appendix (ref).
The logarithm of the asset price is denoted by $x$ and is defined on a filtered space $({\Omega},{\cal F},\mathbb{F}=({\cal F}_t)_{t\geq0})$, equipped with two probability measures, the true statistical probability measure $\mathbb{P}$ and the risk-neutral probability measure $\mathbb{Q}$. We assume that $x$ is an It\^o semimartingale under $\mathbb{Q}$ of the form
where $W$ is a Brownian motion, ${\alpha}$ and ${\sigma}$ denote the drift and volatility of the asset price, respectively, and $J^x$ is the jump part of $x$ (which is given in (ref)).
In this paper, we are interested in the volatility of the volatility process ${\sigma}$ and its components. Assuming that
we develop feasible inference for $({\sigma}^{\sigma}_t)^2+(\overline{\sigma}^{\sigma}_t)^2$ and ${\sigma}_t{\sigma}^{\sigma}_t$. The first of these two quantities is the spot diffusive variance of the process $\sigma$, while ${\sigma}_t{\sigma}^{\sigma}_t$ captures the covariance between diffusive price and volatility moves. In (ref), ${\alpha}^{\sigma}$, ${\sigma}^{\sigma}$ and $\overline{\sigma}^{\sigma}$ are the drift and diffusive coefficients of ${\sigma}$, respectively, $\overline W$ is a Brownian motion independent of $W$, and $J^{\sigma}$ denotes the jump component of ${\sigma}$ (given in (ref)).
Our estimation procedure is based on options written on the underlying asset whose theoretical values are related to the dynamics of $x$ under the risk-neutral probability measure $\mathbb{Q}$, see equation ((ref)) below. In an arbitrage-free setting, the latter is locally equivalent to the statistical probability measure $\mathbb{P}$, and by Girsanov's theorem (see Theorem III.3.24 in JS03), both $x$ and ${\sigma}$ are also It\^o semimartingales under $\mathbb{P}$, with representations of the form (ref) and (ref), respectively. To be notationally precise, one should put superscripts $\mathbb{P}$ or $\mathbb{Q}$ on the drift, the jump part and the Brownian motions in (ref) and (ref). In order to simplify notation, however, we shall refrain from doing so and instead mention the probability measure whenever we refer to (ref) or (ref). Let us also stress at this point that the main quantities we are interested in here, namely, ${\sigma}$, ${\sigma}^{\sigma}$ and $\overline{\sigma}^{\sigma}$, are the same under $\mathbb{P}$ and $\mathbb{Q}$.
As mentioned in the introduction and further explained in Section (ref) below, prices at time $t$ for options with time-to-maturity $T$ allow us to reconstruct the ${\cal F}_t$-conditional characteristic function (under $\mathbb{Q}$) of the normalized price change from $t$ to $t+T$, that is, of
where $u_T=u/\sqrt{T}$ and $\mathbb{E}_t=\mathbb{E}^\mathbb{Q}[\cdot \mid {\cal F}_t]$ denotes ${\cal F}_t$-conditional expectation under $\mathbb{Q}$. Using the fact that ${\cal L}_{t,T}(u)=e^{-\frac12 u^2{\sigma}_t^2} + o(1)$ as $T\downarrow 0$ (under mild assumptions on $x$ and ${\sigma}$), T19 constructs an estimator of the spot variance ${\sigma}_t^2$ by setting
Here, and in the remainder of the paper, we use $o=o_p$ and $O=O_p$ to indicate order in probability. We further add the superscript “uc” as in $O^\mathrm{uc}$ or $o^\mathrm{uc}$ to indicate uniformity in $u\in{\cal U}$, where ${\cal U}$ is an arbitrary compact subset of $(0,\infty)$. In order to remove biases of higher asymptotic order of this estimator, todorov2021bias introduce a bias-corrected version of (ref) by considering a second time-to-maturity
for some $\tau>1$ and defining
The main sources of error of both ${\sigma}^2_{t,T}(u)$ and ${\sigma}^2_{t,T,T'}(u)$ are jump risks in price and volatility and the dynamics of their semimartingale characteristics.
Our strategy of estimating spot volatility of volatility consists of forming sample variance and autocovariance of high-frequency increments of ${\sigma}^2_{t,T}(u)$ and ${\sigma}^2_{t,T,T'}(u)$, that is, of
where with a slight abuse of notation we let
for some $\Delta_n\to0$ and $k_n\to\infty$ with $k_n\Delta_n\to0$. In CT23_a, we have derived higher-order asymptotic expansions for $\Delta^n_i {\sigma}^2_{t,T}(u)$ and $\Delta^n_i {\sigma}^2_{t,T,T'}(u)$, as $\Delta_n\to0$ and $T\to 0$. We will make advantage of these results, which we recall in Appendix (ref), to develop our volatility of volatility and leverage effect estimators.
In practice, one is often interested in estimating volatility or variance of a transform of volatility, that is, of $V_t=F({\sigma}^2_t)$, where $F$ is a $C^2$-function on $(0,\infty)$. Typical functions of interest include $F(x)=x$ (volatility of variance), $F(x)=\sqrt{x}$ (volatility of volatility), $F(x)=\log x$ (volatility of log-variance) and $F(x)= \log \sqrt{x}$ (volatility of log-volatility). For any fixed $0\leq\underline t < t<\overline t<\infty$, since ${\sigma}$ is an It\^o semimartingale, $V$ is again an It\^o semimartingale on $[\underline t, \overline t]$ on the event $\{\inf_{s\in[\underline t,\overline t]}{\sigma}_s^2>0\}$. We are interested in estimating $VV_t$, the spot variance of $V$ at time $t$ and $LV_t$, the spot covariance of $V$ and the asset price. By It\^o's formula, we have
which are the same under $\mathbb{P}$ and $\mathbb{Q}$. As natural but infeasible estimators of $V_t$, we consider
where ${\sigma}^2_{t,T}(u)$ is defined in (ref). Similarly, infeasible approximations of $\Delta^n_i V_t$ are given by
We proceed next with constructing estimators of $V_{t,T}(u)$ and $V_{t,T,T'}(u)$ from options. If $x$ denotes the log-price of an asset, the conditional characteristic function of its increments can be inferred from portfolios of short-dated options using an option-spanning result, see bakshi2000spanning and CM01. More specifically, assuming that the dividend yield associated with $x$ and the risk-free interest rate are both equal to zero (as their effect on short-dated options is negligible), we have
for $u\in\mathbb{R}$, where $O_{t,T}(k)$ denotes the price at time $t$ of an European style out-of-the-money option expiring at $t+T$ and with log-strike of $k$, that is,
In agreement with the notation used so far, $\mathbb{E}_t$ signifies the ${\cal F}_t$-conditional expectation under $\mathbb{Q}$. We remind the reader that $O_{t,T}(k)$ is a put if $k\leq \log(F_{t,T})$ and a call otherwise, where $F_{t,T}$ is the time-$t$ futures price of the asset with expiration date $t+T$.
If the option prices $O_{t,T}(k)$ were continuously observable in $k$, then thanks to (ref), the conditional characteristic function ${\cal L}_{t,T}(u)$, and hence ${\sigma}^2_{t,T}(u)$, would be statistics that one could use to estimate ${\sigma}^2_t$, for example. In practice, there are two complications. First, $O_{t,T}(k)$ is only available on a discrete log-strike grid, say, for
which may be random and vary in $t$ and $T$. We denote the gap between consecutive log-strikes by $\delta_{j,t,T} = k_{j,t,T} - k_{j-1,t,T}$, for $j=2,\dots,N_{t,T}$. Second, the observed option prices contain errors. That is, we only observe
We assume that the errors $\epsilon_{j,t,T}$ are defined on an auxiliary space $(\Omega^{(1)},\mathcal{F}^{(1)})$ equipped with a transition probability $\mathbb{P}^{(1)}(\omega,d\omega^{(1)})$ from $\Omega$, the probability space on which $x$ is defined, to $\Omega^{(1)}$. We further define
Making a simple Riemann sum approximation of the integral in ((ref)) using the available options, a feasible estimator of ${\cal L}_{t,T}(u)$ is now given by
for $u\in\mathbb{R}$. This in turn leads to feasible versions of the estimators from (ref) via
and
For $\widehat{\cal L}_{t,T}(u)$ to be a sufficiently good approximation of ${\cal L}_{t,T}(u)$, we need several assumptions concerning the existence of conditional moments of $x$ under $\mathbb{Q}$, the option observation scheme as well as the observation errors. They are quite similar to those employed in T19 and todorov2021bias. In the following, if expectation is taken under $\mathbb{Q}$, we will not use superscript in the notation; if expectation is under $\mathbb{P}$ or $\overline \mathbb{P}$, we put superscripts to signify this.
We are now ready to introduce our estimators. Under Assumption (ref), we can use the observed option prices to build reasonable estimators of ${\cal L}_{t^n_i,T^n_i}(u)$ and ${\cal L}_{t^n_i,T^{\prime n}_i}(u)$; see T19. If we further have Assumptions (ref) and (ref) stated in the Appendix, Theorem (ref) shows that $\Delta^n_i V_{t,T}(u)$ and $\Delta^n_i V_{t,T,T'}(u)$ provide good approximations of $\Delta^n_i V_t$. These two results combined can be used to construct estimators of $VV_t$ and $LV_t$.
We start with volatility of volatility. Our estimators of $VV_t$ are based on the sample variance and first-order autocovariance of these approximations of $\Delta^n_i V_t$. More precisely, given a truncation function $\tau_n(x)=x\mathbf 1_{\{\lvert x\rvert\leq \upsilon_n\}}$ where $\upsilon_n>0$ is some truncation level, we consider for $u>0$ the following volatility of volatility estimators:
where, recalling the notation in (ref), (ref) and (ref), we define \[ \Delta_i^n\widehat{V}_{t,T}(u)=\widehat V_{t^n_{i-1},T^n_{i-1}}(u)-\widehat V_{t^n_i,T^n_i}(u),\quad \Delta_i^n\widehat{V}_{t,T,T'}(u)=\widehat V_{t^n_{i-1},T^n_{i-1},T^{\prime n}_{i-1}}(u)-\widehat V_{t^n_i,T^n_i,T^{\prime n}_i}(u).\] The reason we include the first-order autocovariance for the volatility of volatility estimators is because this automatically corrects for a bias caused by the option observation errors.
Turning next to the leverage effect, our estimators are based on sample covariances involving the price increments and the increments of the volatility estimates:
Note that here, unlike the case of estimating $VV_t$, we do not need to make corrections for the option observation error as the latter does not introduce an asymptotic bias in $\widehat{LV}^n_{t,T}(u)$ and $\widehat{LV}^n_{t,T,T'}(u)$.
In order to state the theoretical results of this paper, we need one more set of assumptions, which relate $\Delta_n$, $T$, $k_n$, the strike grid size $\delta$ and the truncation threshold $\upsilon_n$ to each other.
We briefly comment on the various rate conditions in the above assumption. The first line of ((ref)) contains conditions that guarantee that the various biases in the recovery of the volatility increment are of higher asymptotic order relative to the rate of convergence in the CLT for the volatility of volatility and leverage effect estimators. The first of them is rather weak when $N$ is high. The second one is due to the biases of exact order $\sqrt{\Delta_n}T$ that arise in an expansion of $\Delta^n_i V_{t,T}(u)$ (see the first line in ((ref))). If we use the bias-corrected estimators based on two maturities, the required condition is weaker than the one needed for estimators based on a single maturity, i.e., $k_nT=O(1)$ versus $k_nT=o(1)$. The third condition requires the length of the estimation window to be not larger asymptotically than the time-to-maturity of the options, which will be the case in applications.
The fourth condition makes sure that $VV_t-\frac{1}{k_n\Delta_n}\int_{t-k_n\Delta_n}^t VV_s ds$ and $LV_t-\frac{1}{k_n\Delta_n}\int_{t-k_n\Delta_n}^t LV_s ds$ (i.e., the biases arising from the difference between spot volatility of volatility / leverage effect and local averages thereof) are asymptotically negligible. Finally, the last condition in the first line of ((ref)) is due to the Riemann approximation error of the integral in ((ref)). This error is typically small for applications such as the one considered in our empirical analysis. Next, the first condition in the second line of ((ref)) is a balance condition between the size of the error in the estimation due to the diffusive component of the process $\sigma^2$ and the one due to the option observation error.\footnote{The rate of convergence of $\widehat{V}_{t,T}(u)$ for a fixed $t$ is $T^{1/4}/\sqrt{\delta}$, see T19. } We note that we allow $\phi$ to take both values of $0$ and $1$, that is, we allow either of these two sources of error to dominate the other one. Finally, the requirements for the threshold $\upsilon_n$ in ((ref)) are mild. Indeed, $\upsilon_n=\infty$ (i.e., no truncation) is permitted.
A feasible CLT for the volatility of volatility estimators is given in the following theorem.
The corresponding CLT result for the leverage effect estimators is given in the next theorem.
As evident from Remark (ref), the rate of convergence for all estimators is $\sqrt{k_n}$ if $\phi>0$, where $\phi$ is the constant in Assumption (ref). If $\phi = 0$, then the rate of convergence of the volatility of volatility estimators slows down to $\sqrt{k_n}\Delta_n/(\delta/\sqrt{T})$ while that of the leverage effect estimators slows down to $\sqrt{k_n}\sqrt{\Delta_n/(\delta/\sqrt{T})}$.\footnote{Even though the volatility of volatility and the leverage effect estimators are not consistent if $\phi=0$ and $\sqrt{k_n}[\Delta_n/(\Delta_n+\delta/\sqrt{T})] \not\to \infty$, the limit results in Theorems (ref) and (ref) continue to hold.} The case $\phi>0$ corresponds to the situation when option observation errors are sufficiently small so that their presence does not affect the rate of convergence of the estimators.\footnote{If $\phi\in(0,1)$, the option observation errors have an effect on the limit variance as evident from Remark (ref).} If this is not the case, i.e., if $\phi=0$, the presence of option observation errors slows down the rate of convergence of the estimators and determines their limit distributions. The feasible CLTs in the above two theorems have the convenient feature, from an applied point of view, that the user does not need to know a priori the value of $\phi$ from (ref) above, i.e., which of the sources of estimation error is asymptotically dominant. Our estimates of the asymptotic variance are constructed in a way that adapts to the situation at hand.
We can compare the rate of convergence of $\widehat{VV}^n_{t,T}(u)$ and $\widehat{LV}^n_{t,T}(u)$ with that of their return-based counterparts. It is easiest to do so in the case when both the underlying asset price and the options written on it are not contaminated by observation errors (microstructure noise). In this case, the rate of convergence of the spot counterparts of the estimators of WM14, vetter2015estimation, AFLWY17 and KX17 for the volatility of volatility and the leverage effect using $k_n$ high-frequency price observations is $k_n^{1/4}$. By contrast, our estimators $\widehat{VV}^n_{t,T}(u)$ and $\widehat{LV}^n_{t,T}(u)$ have a faster rate of convergence of $\sqrt{k_n}$ when there are no option observation errors (or the latter are not too big, i.e., when $\phi >0$). We will see in the Monte Carlo in Section (ref) that the faster rate of convergence of the option-based estimators translates into rather nontrivial efficiency gains in finite samples over their return-based counterparts. This holds true even when the observed option prices contain measurement error, while the observed asset prices do not.
We finish this section with summarizing in Table (ref) existing estimators related to the estimation of volatility of volatility and the leverage effect. We only list nonparametric estimators with a feasible CLT in the table. We do not include the leverage effect estimators of andersen2015exploring and KX17 based on the VIX index because they do not estimate $LV_t$ in general, which is what we are after. This is further discussed in Section (ref) below. The first four entries in the table are estimators of spot volatility based on return or option data. The volatility of volatility and leverage effect estimators from high-frequency asset price data given in the table are obtained by integrating over time the spot quantities examined in this paper. It is relatively straightforward to extend these results to the case of spot volatility of volatility and spot leverage effect estimation. In addition, some of the existing leverage effect estimators are estimators of the correlation--and not covariance--between the diffusive price and the volatility. Again, these results can be extended to the estimation of the leverage effect in the way we define it here.
In this section, we evaluate the performance of the proposed volatility of volatility and leverage effect estimators on simulated data.
We use the following model for the underlying asset price $X_t=e^{x_t}$, under the risk-neutral probability measure $\mathbb{Q}$, to generate the true option prices:
where
and $W_t$ and $B_t$ are $\mathbb{Q}$-Brownian motions with $\textrm{corr}(dW_t,dB_t) = \rho dt$, and $\mu$ is an integer-valued random measure with $\mathbb{Q}$-compensator $dt\otimes \nu_t(dx)$ and
In the above specification for $X$, the stochastic variance is modeled as a square-root diffusion process like in the popular Heston model heston. The price jumps have intensity that is affine in the level of diffusive variance like in DPS00 and subsequent empirical option pricing work. Our jump specification is a time-changed double-exponential model, with the time-change being the integrated diffusive variance.
We consider three parameter settings for the above model. The parameter values for the three cases are given in Table (ref). In all of them, the unconditional mean of the variance is $\theta_v = 0.02$. In the first specification, the volatility is very persistent with half-life of a shock to stochastic variance equal to six months. In the second and third specifications, the half-life of a shock to variance is one month and ten business days, respectively. In all cases, the parameter $\rho$ is set to $-0.9$ implying strong negative correlation between price and variance diffusive shocks. We note that the Feller condition ($\sigma_v^2\leq 2\kappa_v\theta_v$) puts an upper bound on $\sigma_v$ and this means that for more persistent dynamics the volatility of volatility is smaller. Turning next to the jump specification, we set $\lambda_-=50$ and $\lambda_+=100$. This choice implies tail decays of out-of-the-money puts and calls similar to those of observed options written on the S&P 500 index, see e.g., AFT,AFT_b. Finally, we set $c_{\pm}$ according to \[c_- = 0.9\times \frac{\lambda_-^{3}}{2}~\textrm{and}~c_+ = 0.1\times \frac{\lambda_+^{3}}{2},\] which implies that spot jump variation is equal to spot diffusive variance, and further that $90\%$ of the jump variation is due to negative jumps. This separation of the risk-neutral variation into a diffusive part and one due to positive and negative jumps is similar to that implied from parametric models fitted to observed S&P 500 index options, see e.g., AFT_b.
For simplicity, the dynamics of $x$ under $\mathbb{P}$ is the same as that under $\mathbb{Q}$ with one exception. Mainly, we do not allow for jumps under $\mathbb{P}$. A more realistic specification would be one in which we allow for price jumps but with much smaller size than the one they have under $\mathbb{Q}$. We do not consider such an extension of the setup as it has only negligible effect on the results.
Observed options are given by
where $\{z_{t,T}(j)\}_{j=1}^{N_{t,T}}$ are sequences of i.i.d.\ standard normal variables which are independent of each other. The size of the observation error is calibrated to roughly match bid--ask spreads of index option data. We set $X_0 = 2500$ and $\Delta_n = 1/(252\times 80)$, which corresponds approximately to sampling option data every five minutes in a 6.5 hours trading day. We note that our unit of time is one year and we adopt a business time convention in which one business day is of length $1/252$. We set $k_n = 80$ which means that we use all intraday option data in the analysis. At each point in time, the strikes are multiples of $5$. The strikes below and above the current price are extended in both directions by increments of $5$ until the true out-of-the-money option price falls below $0.075$. This specification of the strike grid mimics that of available S&P 500 index options. Next, the value of the variance at the beginning of the local time window, $V_0$, is set to the 25th, 50th or 75th quantile of its marginal distribution. Finally, the short and long tenor of the options at the beginning of the time window are set to $T= 3/252$ and $T' = 6/252$, which correspond to $3$ and $6$ business days to expiration, respectively.
To reduce the Riemann sum approximation error of the integral in ((ref)), we compute option prices on a strike grid with mesh $2.5$ from the observed ones using linear interpolation in Black--Scholes Implied Volatility (BSIV) space (recall that the strikes of observed options are multiples of $5$).
For the computation of the volatility estimators, we need to choose the value of the characteristic exponent, i.e., the value of $u$ in $V_{t,T}(u)$. We do this in a data-driven way using the options at the first time point in the local window by setting $u$ to
where $\overline{u}_{t^n_{k_n},T^n_{k_n}} = \sqrt{-2\log(0.05)}/\widehat{\sigma}_{t^n_{k_n},ATM}$ and $\widehat{\sigma}_{t^n_{k_n},ATM}$ is the at-the-money Black--Scholes implied volatility at time $t^n_{k_n}$ for the shortest available maturity on that day. We thus look at $\widehat{V}_{t^n_i,T^n_i}(\widehat{u}_{t^n_{k_n},T^n_{k_n}})$ and $\widehat{V}_{t^n_i,T^n_{i},T^{\prime n}_{i}}(\widehat{u}_{t^n_{k_n},T^{\prime n}_{k_n}})$. Finally, since the model does not feature volatility jumps, we do not perform truncation, i.e., we set $\upsilon_n = \infty$.
We compare the performance of our option-based estimators with their counterparts formed from high-frequency returns. We assume that we sample returns at a higher frequency than we sample option prices at. Using the higher sampling frequency, we then form estimates of volatility at the times $t_i^n$. More specifically, denote
for some sequence $l_n\rightarrow\infty$. Using the above spot volatility estimates, we have the following return-based volatility of volatility and leverage effect estimators:
We can further de-bias $\widehat{VV}^n_{t,ret}(u)$ to account for the effect from the nonlinear transformation of $\widehat{\mathcal{L}}_{t_i^n,ret}(u)$ in forming the statistic (no such de-biasing is needed for $\widehat{LV}^n_{t,T}(u)$). We do not do this for simplicity. We note also that we could have used alternatively local truncated volatility in constructing $\widehat{VV}^n_{t,ret}(u)$ and $\widehat{LV}^n_{t,ret}(u)$. We prefer the current estimators as they are the direct counterparts to the option-based ones.
As for the option-based estimators, we set $u$ in a data-driven way. Towards this end, define
and set
With this choice of $u$, our return-based estimates become $\widehat{VV}^n_{t,ret}(\widehat{u}_{t,ret})$ and $\widehat{LV}^n_{t,ret}(\widehat{u}_{t,ret})$. We implement these estimators with $l_n = 72$. This corresponds to sampling every five seconds and forming local volatility estimates over blocks of length five minutes.
The Monte Carlo results for estimating variance of log-variance (i.e., $VV_t$ in (ref) with $F=\log x$) are reported in Table (ref). We can draw several conclusions from them. First, the estimators which use options with one tenor only tend to be downward biased except for the very persistent case S. The size of this negative bias in most configurations increases with time to maturity of the options. This can be explained intuitively with higher-order biases present in the option-based spot volatility estimator, which increase in size as a function of the time to maturity. For example, the volatility of volatility generates a downward bias in the spot volatility estimator. This bias is naturally bigger for the third volatility specification (case F). As a result, the bias in our volatility of volatility estimators, using options with one tenor only, tends to be larger in magnitude for this volatility specification (reported in Panel C of the table). Second, the volatility estimator using two tenors corrects for the biases in spot volatility due to mean reversion in volatility and volatility of volatility. As a result, the estimator $\widehat{VV}_{0,T,T'}$ tends to have much smaller in magnitude biases in cases M and F. The reduction in bias is particularly large for the most volatile specification (case F). The cost of the reduction in bias in $\widehat{VV}_{0,T,T'}$ is increased volatility of that estimator when compared with the single-tenor estimators $\widehat{VV}_{0,T}$ and $\widehat{VV}_{0,T'}$.
Third, volatility of volatility is most difficult to estimate when it is smallest, i.e., in scenario S. The reason for this is because in this case the signal to noise ratio is at its lowest. As a result, various biases in the estimation play a more important role in a relative sense now. In such a situation, the two tenor option-based estimator has comparable bias as its one-tenor counterparts. Finally, comparing the option-based estimates with the return-based ones, we can see the vast improvement offered by the option data for measuring volatility of volatility. This is true regardless of the volatility specification and the starting value of volatility. Recall that our return-based estimator uses much higher sampling frequency than the option-based estimators. In spite of that, the option-based estimators are much more precise. This illustrates the advantages of using option data for inference related to volatility.\footnote{The performance of the return-based estimator will worsen further if one is to allow for market microstructure noise in the observed asset price.}
We proceed next with a discussion of the Monte Carlo results for estimating the leverage effect corresponding to the log-variance (i.e., of $LV_t$ in (ref) with $F=\log x$). These results are reported in Table (ref). We note that for the Heston volatility model used in the Monte Carlo, $LV_t$ is a constant equal to $\rho\times\sigma_v$. Qualitatively, the results for estimating $LV_t$ are similar to those for estimating $VV_t$. In particular, the biases in the estimation are larger in magnitude for the fastest mean reversion simulation scenario F. The use of two tenors is rather beneficial in terms of reducing the downward bias in the estimation. Comparing the results in Tables (ref) and (ref), we can see that the precision of the estimation of the leverage effect is higher than that of the volatility of volatility. Finally, the option-based estimators again perform much better than the return-based one. The advantage in relative sense is not as big as in the case of volatility of volatility but is nevertheless quite nontrivial.
We finish this section with a comparison of the estimates of volatility of volatility and the leverage effect when one uses the VIX index as a proxy for the spot volatility as done by andersen2015exploring and KX17. The theoretical value of the squared VIX index is $\frac{1}{\mathcal{T}}\mathbb{E}_t\Bigl[\int_t^{t+\mathcal{T}}\sigma_s^2ds + 2\int_t^{t+\mathcal{T}}\int_{\mathbb{R}}(e^z-1-z)\mu(ds,dz)\Bigr]$. For the model used in the Monte Carlo, the value of the square of the VIX index is given by
This implies that the ratio of the diffusion coefficients of $\log(VIX_t^2)$ and $\log(\sigma_t^2)$ in the model is given by \[ s_t = \frac{2V_t}{VIX_t^2}\frac{1-e^{-\kappa_v\mathcal{T}}}{\kappa_v\mathcal{T}}. \] Therefore, the counterpart of $VV_t$ for $VIX_t$ is $s_t^2\times VV_t$ and of $LV_t$ is $s_t\times LV_t$. As an example, the ratio $s_t$ takes values in the range $(0.3,\,0.6)$ for scenario F, depending on the starting value of the volatility considered here. This implies large differences in volatility of volatility and the leverage effect depending on whether one uses $\sigma_t$ or $VIX_t$. Note also that this gap is time-varying implying that the dynamics of the series will be different as well.
We apply the developed estimation techniques to study the volatility of volatility and leverage effect of the S&P 500 market index. We use high-frequency data of S&P 500 index options traded on the CBOE options exchange for the period 2016--2020. On each trading day in our sample period, we record option prices every five minutes, starting from 9.35 AM EST until 4.00 PM EST. At each point in time we record out-of-the-money option mid-quotes, keeping only strikes with nonzero bids and with a ratio of ask/bid of less than $10$. The moneyness of the options is determined on the basis of a synthetic futures extracted from the option data using put-call parity and short-term T-bill interest rate. We remove from the analysis maturities for which the gap between strikes of the three nearest-the-money puts and calls is above $5$ (which is the minimum strike gap for SPX options). We remove also maturities for which $\max \{\widehat{O}_{t,T}(k_{t,T}(1)),\widehat{O}_{t,T}(k_{t,T}(N_{t,T}))\}/\max_{j=1,...,N_{t,T}}\widehat{O}_{t,T}(k_{t,T}(j))>0.025$.
From the remaining maturities on a given trading day, we keep the two closest to expiration with time to maturity between $2$ and $16$ business days. We require that the gap between the two tenors be at least three business days. The median time-to-maturities $T$ and $T'$ at the end of each trading day in our sample are approximately $2$ and $6$ business days (and in $95\%$ of the days in the sample we use only options that have less than two weeks to expiration). These numbers are very close to the ones used in the Monte Carlo. In choosing $T$ and $T'$, there is a bias-variance trade-off. Smaller $T$ and $T'$ imply less biased spot volatility estimators (as our asymptotic expansions are for $T,T'\rightarrow 0$) but more noisy ones as we have fewer option observations per maturity at each point in time as $\delta$ (the mesh of the strike grid) is fixed. Our Monte Carlo experiments showed that, for $T$ and $T'$ similar to the ones used in the empirical application, the finite sample performance of the estimators is good.
Finally, we exclude from the analysis February 5 and 6, 2018, during which there were wild intraday movements in volatility associated with the collapse of inverse exchange-traded volatility instruments. The choice of the characteristic exponent $u$ is done exactly as in our simulation expirement. As in the Monte Carlo, we compute option prices on an equidistant strike grid with strike gap of $2.5$ via linear interpolation in BSIV space from the observed ones. We perform truncation of the volatility increments by setting the truncation parameter $\upsilon_n$ to $$\widehat{\upsilon}_n = 3\left(\frac{\pi}{8}\frac{1}{(k_n-1)\Delta_n}\sum_{l=0}^{3}\sum_{i=2}^{k_n}|\Delta_i^n\widehat{V}_{t-l/252}|\,|\Delta_{i-1}^n\widehat{V}_{t-l/252}|\right)^{0.49},$$ where $\widehat{V}_t$ is one of the three considered spot variance estimators.
In the middle panel of Figure (ref), we plot our estimates of the market volatility of volatility, i.e., our estimates of the square of the diffusive coefficient of $\log(\sigma_t^2)$. In the top panel of the figure, we also plot the market spot diffusive volatility. We display $5$-day moving averages of the daily estimates in order to focus on lower frequency variation in the time series and minimize the impact of measurement error. As seen from the figure, market volatility of volatility displays only mild variation during our sample with occasional and relatively short-lived spikes. We plot both $\widehat{VV}_{t,T}$ as well as $\widehat{VV}_{t,T,T'}$. Exactly as in the Monte Carlo, $\widehat{VV}_{t,T,T'}$ is higher than $\widehat{VV}_{t,T}$. The gap between the two series is particularly large during 2017 when volatility was relatively low. Recall from the Monte Carlo that when volatility is low (relative to its mean), the downward bias in $\widehat{VV}_{t,T}$ tends to be more significant.
Related to that, the reported volatility of spot volatility estimates are somewhat larger than the values of the VVIX index during the same period (when reported in the same units as our estimators). The reason for this is that VVIX is a measure of volatility of the VIX index, which is a conditional expectation of one-month future volatility. The mean-reversion in volatility makes the VIX smoother than the spot volatility and this can explain the differences between our estimates and the VVIX, see equation ((ref)) above and the discussion afterwards.
Consistent with earlier work cited in the introduction that argues for a partial disconnect between volatility risk and volatility of volatility risk, we find weak correlation between $\widehat{V}_{t,T}(u)$ and $\widehat{VV}_{t,T,T'}$. For example, the big increase in market volatility in the Spring of 2020 is not accompanied by a significant change in the volatility of volatility (recall that $\widehat{VV}_{t,T,T'}$ is an estimate of volatility of log-variance).
In the bottom panel of Figure (ref), we plot our estimates of a $5$-day moving average of the market leverage effect. Unlike the case of volatility of volatility, the two estimates $\widehat{LV}_{t,T}$ and $\widehat{LV}_{t,T,T'}$ are now much closer on average. We note also that the leverage effect increases sharply in magnitude at the onset of the pandemic in the Spring of 2020 when the market volatility also increased a lot.
As mentioned in the introduction, our nonparametric estimates for the volatility of volatility and the leverage effect can be used as diagnostic tools for the specification of volatility dynamics. To illustrate this, we can contrast the time series properties of our nonparametric estimates with those implied by two popular volatility specifications considered in prior work. One is the Heston model that we used in our Monte Carlo, with the possible addition of volatility jumps, see e.g., DPS00. The other is an exponential Ornstein--Uhlenbeck (OU) volatility specification, see e.g., chernov2003alternative, in which the logarithm of the diffusive variance follows an Ornstein--Uhlenbeck process.
The Heston model plus jumps and the exponential Ornstein--Uhlenbeck (OU) model differ in terms of the volatility of volatility and leverage effect they generate. For the Heston model, $VV_t$ is proportional to $1/\sigma_t^2$ while $LV_t$ is constant. On the other hand, for the exponential-OU volatility model, $VV_t$ is constant while $LV_t$ is proportional to $\sigma_t$. Our nonparametric evidence shows that the exponential-OU volatility model can better rationalize the observed dynamics of $VV_t$ and $LV_t$. In particular, the estimated volatility of volatility exhibits little time series variation while the estimated leverage effect is inversely related to the market volatility. These features of the observed estimates are at odds with those implied by the classical Heston model.
The volatility of volatility and leverage effect are asset price characteristics that are notoriously difficult to estimate from high-frequency price data, particularly when price and volatility have jumps and market microstructure noise is present. This is mainly due to the latency of volatility and the fact that return-based spot volatility estimates are rather noisy.
In this paper, we propose to estimate the volatility of volatility and the leverage effect using high-frequency data from short-dated options written on the underlying asset. Our nonparametric estimators are consistent, asymptotically mixed normal and permit feasible inference in the joint presence of price and volatility jumps and observation errors. A simulation study reveals nontrivial efficiency gains from using the proposed estimators over ones constructed using only high-frequency price data.
Applying our estimators to S&P 500 index high-frequency option data, we generate reliable market spot volatility of volatility and leverage effect estimates using as little as one day of $5$-minute data. Our estimates provide nonparametric evidence in favor of an exponential-OU model and against a Heston-type model for market volatility.